Common Distribution Function Research Articles

Weighted Approximations of Tail Copula Processes with Application to Testing the Multivariate Extreme Value Condition

Consider n i.i.d. random vectors on R2, with unknown, common distribution function F. Under a sharpening of the extreme value condition on F, we derive a weighted approximation of the corresponding tail copula process. Then we construct a test to check whether the extreme value condition holds by comparing two estimators of the limiting extreme value distribution, one obtained from the tail copula process and the other obtained by first estimating the spectral measure which is then used as a building block for the limiting extreme value distribution. We derive the limiting distribution of the test statistic from the aforementioned weighted approximation. This limiting distribution contains unknown functional parameters. Therefore, we show that a version with estimated parameters converges weakly to the true limiting distribution. Based on this result, the finite sample properties of our testing procedure are investigated through a simulation study. A real data application is also presented.

On the asymptotic distribution of certain bivariate reinsurance treaties

Let { ( X n , Y n ) , n ≥ 1 } be bivariate random claim sizes with common distribution function F and let { N ( t ) , t ≥ 0 } be a stochastic process which counts the number of claims that occur in the time interval [ 0 , t ] , t ≥ 0 . In this paper we derive the joint asymptotic distribution of randomly indexed order statistics of the random sample ( X 1 , Y 1 ) , ( X 2 , Y 2 ) , … , ( X N ( t ) , Y N ( t ) ) which is then used to obtain asymptotic representations for the joint distribution of two generalised largest claims reinsurance treaties available under specific insurance settings. As a by-product we obtain a stochastic representation of a m -dimensional Λ -extremal variate in terms of iid unit exponential random variables.

The Mean Residual Life Function of a<tex>$k$</tex>-out-of-<tex>$n$</tex>Structure at the System Level

In the study of the reliability of technical systems, k-out-of-n systems play an important role. In the present paper, we consider a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that, at time t, all the components of the system are working, we propose a new definition for the mean residual life (MRL) function of the system, and obtain several properties of that system.

Law of iterated logarithm and almost sure limit points for random sums

Let { X n , n ≥ 1 } be a sequence of independent identically distributed (i.i.d) random variables with a common distribution function F . When F belongs to the domain of partial attraction of a Semi-Stable law with index α , 0 < α < 2 , we give complete solution to the results of R. Vasudeva and G. Divanji [Law of iterated logarithm for random subsequences, Statist. Probab. Lett. 12 (1991) 189–194], where they obtained Chover’s form of the law of iterated logarithm for random subsequences. Further, we extended the situation in obtaining almost sure limit points for random subsequences.

Review: An investigation of single particle breakage tests for coal handling system of the gladstone port

Aspects of the literature on single particle breakage test have been reviewed in this article. The test procedures that are commonly used by the researchers in examining and measuring the breakage characteristics of the ore and coal particles are also discussed. It appears that most of the common size distribution function fitting techniques were not suitable for accurate representation of the size distributions obtained from a pendulum breaking process. The single impact test, double impact test (drop weight test, pendulum test) and slow compression test can be used to study the behaviour of the single particle breakage events. The single impact test, slow compression test and drop weight test cannot measure the energy utilization pattern in single particle breakage events, but this can be determined from the pendulum test. The energy utilized for breakage was predominantly dependent upon the size and shape of the specimen, level of input energy and the breakage properties of the specimen. This review highlights that the size distribution curves were linear in the fine particle region and have varying curvature in the coarser region, the gradient of the linear fine particle region of the size distribution curves increases with an increase in the specific comminution energy. The comminution energy increases with input energy at lower levels of input energy but at the higher levels of input energy the comminution energy did not show the same proportional increase. At a given level of input energy, the size distribution resulting from the breakage of the particles by the pendulum apparatus can be represented by a one-parameter family of curves.

Estimation of the truncation probability with left-truncated and right-censored data

Let (, , ) be i.i.d. random vectors such that (, ) is independent of and . Let F, Q, and G denote the common distribution function of , , and , respectively. For left-truncated and right-censored data, one can observe nothing if and observe ( ), with and , if . A problem of interest is the estimation of the truncation probability . Under the constraint that , Wang [Wang, M.-C., 1991, Nonparametric estimation from cross-sectional survival data. Journal of Americal Statistical Association, 86, 130–143.] suggested estimating α by α n = ∈ t [1−F n (s−)]dG n (s), where F n and G n are non-parametric maximum likelihood estimate of the distributions F and G, respectively. In this note, using inverse-probability-weighted estimators of F, G, and Q, we obtain two alternative representations, ˜α n and αˆ n , for α n . Simulation study shows that the estimator α n works satisfactorily for moderate sample size.

Multivariate weighted renewal functions

Let ( X , Y ) , ( X 1 , Y 1 ) , ( X 2 , Y 2 ) , … denote independent positive random vectors with common distribution function F ( x , y ) = P ( X ⩽ x , Y ⩽ y ) with F ( x , y ) < 1 for all x , y . Based on the X i and the Y j we construct the sum sequences S n 1 and S m 2 respectively. For a double sequence of weighting constants { b ( n , m ) } we associate a weighted renewal function G ( x , y ) defined as G ( x , y ) = ∑ n = 0 ∞ ∑ m = 0 ∞ b ( n , m ) P ( S n 1 ⩽ x , S m 2 ⩽ y ) . The function G ( x , y ) can be expressed in terms of well-known renewal quantities. The main goal of this paper is to study asymptotic properties of G ( x , y ) . In the one-dimensional case such results have been obtained among others by Omey and Teugels [Weighted renewal functions: a hierarchical approach, Adv. in Appl. Probab. 34 (2002) 394–415.] and Alsmeyer [Some relations between harmonic renewal measures and certain first passage times, Statist. Probab. Letters 12 (1991) 19–27; On generalized renewal measures and certain first passage times, Ann. Probab. 20 (1992) 1229–1247]. Here we prove a multivariate version of the elementary renewal theorem and moreover we obtain a rate of convergence result in this elementary renewal theorem. We close the paper with an application and some concluding remarks. For convenience we prove and formulate the results in the two-dimensional case only.

Multiple maxima in multivariate samples

Let { X n , n ⩾ 1 } be a sequence of independent random vectors in R d , d ⩾ 2 , with common continuous distribution function F. For fixed n ⩾ 2 , we say that a multiple maximum occurs among the sample points X 1 , … , X n if M n = X i for some i = 1 , … , n , with M n the componentwise sample maximum. In this paper, we investigate the asymptotic behaviour ( n → ∞ ) of the probability of observing a multiple maximum by imposing an asymptotic tail condition on F.

Iterative Estimation of the Extreme Value Index

Let {X n , n ≥ 1} be a sequence of independent random variables with common continuous distribution function F having finite and unknown upper endpoint. A new iterative estimation procedure for the extreme value index γ is proposed and one implemented iterative estimator is investigated in detail, which is asymptotically as good as the uniform minimum varianced unbiased estimator in an ideal model. Moreover, the superiority of the iterative estimator over its non iterated counterpart in the non asymptotic case is shown in a simulation study.

Some limit theorems for large deviations of sums of independent random variables with a common distribution function from the domain of attraction of the normal law

We investigate some aspects of asymptotic behavior of the probabilities P (Sn ≥ α bn), where Sn is a sum of n independent random variables with common distribution function from the domain of attraction of the normal law, α is a positive number, and bn tends to infinity. In particular, necessary and sufficient conditions are given under which the series \(\sum\limits_n {fnP\left( {Sn \geqslant \alpha bn} \right)} \) converges or, after some normalization, has a limit as α ↘ α0, where α0 is a positive constant and fn is some positive sequence of a rather general form. Bibliography: 14 titles.

Estimation of Tails and Related Quantities Using the Number of Near-Extremes

Abstract In an insurance context, consider {Xn, n ≥ 1} random claim sizes with common distribution function F and {N(t), t ≥ 0} an integer valued stochastic process tfiat counts the number of claims occurring during the time interval [0, t]. Based on the number of near-extremes which are the observations Xi near the largest or the mth largest observation, we derive in this article a strongly consistent estimator of upper tails of X1. Furthermore, estimators for both the tail index and the upper endpoint are introduced when F is a generalized Pareto distribution. Asymptotic normal law for the proposed estimators is additionally presented.

On Exact Asymptotics in the Weak Law of Large Numbers for Sums of Independent Random Variables with a Common Distribution Function from the Domain of Attraction of a Stable Law. II

Let us consider independent identically distributed random variables $X_1, X_2, \dots\,$, such that $$ U_n=\frac{S_n}{B_n} -n\,a_n \longrightarrow \xi_\alpha\qquad \mbox{weakly as}\quad n\to\infty, $$ where $S_n = X_1 + \cdots + X_n$, $B_n>0$, $a_n$ are some numbers $(n\geq 1)$, and a random variable~$\xi_\alpha$ has a stable distribution with characteristic exponent~$\alpha\in[1,2]$. Our basic purpose is to find conditions under which $$ \sum_n f_n{\bf P}\big\{U_n\geq\varepsilon\varphi_n\big\}\sim \sum_n f_n{\bf P}\big\{\xi_\alpha\ge\varepsilon\varphi_n\big\}, \qquad\varepsilon\searrow 0, $$ with a positive sequence $\varphi_n$, which tends to infinity and satisfies mild additional restrictions, and with a nonnegative sequence $f_n$ such that $\sum_n f_n =\infty $.

Estimation of Tails and Related Quantities Using the Number of Near-Extremes

In an insurance context, consider {Xn, n ≥ 1} random claim sizes with common distribution function F and {N(t), t ≥ 0} an integer valued stochastic process tfiat counts the number of claims occurring during the time interval [0, t]. Based on the number of near-extremes which are the observations Xi near the largest or the mth largest observation, we derive in this article a strongly consistent estimator of upper tails of X1. Furthermore, estimators for both the tail index and the upper endpoint are introduced when F is a generalized Pareto distribution. Asymptotic normal law for the proposed estimators is additionally presented.

On the mean past lifetime of the components of a parallel system

In the study of reliability of the technical systems and subsystems, parallel systems play a very important role. In the present paper, we consider a parallel system consisting of n identical components with independent lifetimes having a common distribution function F. It is assumed that at time t the system has failed. Under these conditions, we obtain the mean past lifetime (MPL) of the components of the system. Some properties of MPL are studied. It is shown that the underlying distribution function F can be recovered from the proposed MPL. Also, a comparison between two parallel systems are made based on their MPLs in the case where the components of the system are ordered in terms of reversed hazard rate. Finally a characterization of the uniform distribution is given based on MPL.

Almost Sure Limit Points for Subsequences of Partial Sums

Let { Xn, n ≥ 1} be a sequence of i. i. d. random variables with a common distribution function F. When F belongs to the domain of partial attraction of a Semi Stable law with index α, 0 &lt; α &lt; 2, we obtain almost sure limit points for subsequences of partial sums, along with some boundary crossing problems.

Precise large deviations for sums of random variables with consistently varying tails

Let {Xk, k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation μ &gt; 0. Under the assumption that the tail probability is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums Sn and the random sums SN(t), where N(·) is a counting process independent of the sequence {Xk, k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

Precise large deviations for sums of random variables with consistently varying tails

Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation μ &amp;gt; 0. Under the assumption that the tail probability is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N(t), where N(·) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

Asymptotic behavior of tail and local probabilities for sums of subexponential random variables

Let {X k , k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (−∞, ∞), and let τ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {X k , k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· &amp;gt; x) and the local probabilities P(x &amp;lt; · ≤ x + h) of the quantities and for n ≥ 1, and their randomized versions X (τ), S τ and S (τ), where X 0 = 0 by convention and h &amp;gt; 0 is arbitrarily fixed.

Asymptotic behavior of tail and local probabilities for sums of subexponential random variables

Let {Xk, k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (−∞, ∞), and let τ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {Xk, k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· &gt; x) and the local probabilities P(x &lt; · ≤ x + h) of the quantities and for n ≥ 1, and their randomized versions X(τ), Sτ and S(τ), where X0 = 0 by convention and h &gt; 0 is arbitrarily fixed.

On Exact Asymptotics in the Weak Law of Large Numbers for Sums of Independent Random Variables with a Common Distribution Function from the Domain of Attraction of a Stable Law

Let us consider independent identically distributed random variables $X_1, X_2, \dots\,$, such that $$ U_n=\frac{S_n}{B_n} -n\,a_n \longrightarrow \xi_\alpha\qq \mbox{weakly as}\quad n\to\infty, $$ where $S_n = X_1 + \cdots + X_n$, $B_n>0$, $a_n$ are some numbers $(n\geq 1)$, and a random variable $\xi_\alpha$ has a stable distribution with characteristic exponent $\alpha\in (0, 2)$. Our basic purpose is to find conditions under which $$ \sum_n f_n {\bf P}\{|U_n|\geq\varepsilon\varphi_n\}\sim \sum_n f_n {\bf P}\{|\xi_\al|\ge\varepsilon\varphi_n\},\qquad\varepsilon\searrow 0, $$ with a positive sequence $\varphi_n$, which tends to infinity and satisfies mild additional restrictions, and with a nonnegative sequence $f_n$ such that $\sum_n f_n =\infty $.